Translations: generalizing relative expressiveness between logics
Abstract
There is a strong demand for precise means for the comparison of logics in terms of expressiveness both from theoretical and from application areas. The aim of this paper is to propose a sufficiently general and reasonable formal criterion for expressiveness, so as to apply not only to modeltheoretic logics, but also to Tarskian and prooftheoretic logics. For modeltheoretic logics there is a standard framework of relative expressiveness, based on the capacity of characterizing structures, and a straightforward formal criterion issuing from it. The problem is that it only allows the comparison of those logics defined within the same class of models. The urge for a broader framework of expressiveness is not new. Nevertheless, the enterprise is complex and a reasonable modeltheoretic formal criterion is still wanting. Recently there appeared two criteria in this wider framework, one from GarcíaMatos & Väänänen and other from L. Kuijer. We argue that they are not adequate. Their limitations are analysed and we propose to move to an even broader framework lacking modeltheoretic notions, which we call “translational expressiveness”. There is already a criterion in this later framework by Mossakowski et al., however it turned out to be too lax. We propose some adequacy criteria for expressiveness and a formal criterion of translational expressiveness complying with them is given.
Contents:
 1 Introduction
 2 Multiclass expressiveness
 3 Translational expressiveness: obtaining a still wider notion of expressiveness
 4 Conclusions
1 Introduction
It is very common for those who work with logic to make comparisons such as “the logic is more expressive than ”, “ is stronger than ”, “ is included in ”, “ can be reduced to ”, etc. Such assertions are often made on imprecise grounds and, though possibly being nonambiguous and nonproblematic, the lack of clarity around the usage of these concepts can generate terminological confusion across the literature (e.g. [Hum05]) and harden the comparison of formal results.
In the literature, the notion of logic inclusion or sublogic (these terms will be used interchangeably here) is pretty much linked with language and axiomatic extensions, which on their turn are linked with “strength”, that is, the capacity of proving theorems or having valid formulas. Now the concept of sublogic is sometimes associated with strength and sometimes associated with expressiveness, and sometimes with both (e.g. in [Béz99]), which is known to be the case of paradoxes [MDT09]. Three kinds of systems are relevant here: modeltheoretic logics, Tarskian and prooftheoretic logics, they will now be briefly defined. A logic is called modeltheoretic if it is defined semantically and presented as a sequence (), where is a set of formulas, is a class of models and is a satisfaction relation on . A logic is Tarskian if it is defined as (), where is a consequence relation on (possibly multiconsequence). Finally, is a prooftheoretic logic if it is defined as , where is a set of inference rules.
In modeltheoretic logics there is a straightforward approach to expressiveness that is also reasonably taken as a definition of logic inclusion: a logic is at least as expressive/includes if every class of structures characterizable in is also characterizable in (see e.g. [Lin74, p. 129] and [BF85]). This naturally only holds for logics defined within the same class of structures. If one wants also to compare logics defined within different classes of structures, then it does not seem adequate to use the concept of sublogic, as we shall see below. It is better to use the concept of expressiveness.
There is no straightforward approach to expressiveness for Tarskian and prooftheoretic logics (TPL, for short). As for sublogic, in TPL it is also linked with language and axiomatic extensions. However, we can often see “sublogic” relations taken in a wider sense, i.e. when, for two given logics and , it happens that is not a language/axiomatic extension of , but there is a certain mapping of formulas into formulas respecting the consequence relation. These cases are normally interpreted as saying that is included/embeddable/reconstructible/interpretable/can be simulated in . We propose to call these as expressiveness relations whenever they can be seen as modeling the following intuition
For every sentence , there is an sentence with the same meaning.
This same intuitive explanation of expressiveness holds for modeltheoretic logics, and is used as a basis for formal criteria therein (e.g. [BF85, p. 42]). Thus we can have a reasonably homogeneous concept for comparing logics: that of expressiveness. We shall reserve the term “sublogic” just when there are axiomatic or language extensions, and we shall not use the term “strength” because it is ambiguous between expressive and deductive strength.
A precise definition for the notion of relative expressiveness for modeltheoretic logics was given already in the 1970s (e.g. in [Lin74] and [Bar74]). As we said, this definition is based on the capacity of characterizing structures and underlies each of the socalled Lindströmtype theorems,
Singleclass expressiveness Considering modeltheoretic logics defined within the same class of structures, the above intuition can be captured easily since there is a common ground where sentences can be compared. This common ground is easily achieved by defining the meaning of a sentence in a logic as (, for short). Thus we call this framework singleclass expressiveness. Since every sentence in is mapped to a sentence in having the same meaning, this framework of expressiveness can be seen as consisting of certain formulamappings between modeltheoretic logics. A formal definition for it is then straightforward. Let be a signature and let and be modeltheoretic logics.
Definition 1.1 ().
is at least as expressive as () if and only if (iff, for short) for every sentence there is a sentence such that .
Notice that here the class of models is the same for both and , and share the same nonlogical symbols. The above definition can be paraphrased in terms of elementary classes:
Despite being the basis for many important results, is very limited. It is not only restricted to modeltheoretic logics, but it requires the classes of structures being compared to share the same signature. As a consequence, it only allows the comparison of logics defined within the same class of structures. The urge for a broader definition is not new.
Even among those expressiveness results using , we can notice some flexibility in its application. One such example appears in [AFFM11], where the definition of above is given, but afterwards (p. 307) it is informally relaxed in order to allow changes of signature, thus the proper definition being used appears to be the one based on projective classes (). The problem is that elsewhere we get different results depending on whether we use or , as Shapiro showed [Sha91, p. 232]: and , but and .
Remaining within modeltheoretic logics, a wider framework —let us call it multiclass— would comprise besides formulamappings also structuremappings, thus allowing structures of one logic to be mapped to structures of the other. This would enable the comparison of logics defined within different classes of structures. Recently there appeared two formal definitions of multiclass expressiveness, to wit [GMV07] and [Kui14]. In the sequence we will present them and argue that they are not adequate.
There have been also early claims outside abstract modeltheory relating logics in the sense of (E) above, but no explicit definitions of the main concepts involved were given. Gödel used his result on the interpretation of classical into intuitionistic logic to infer that, contrary to the appearances, it is classical logic that is contained in intuitionistic logic [Göd01, p. 295]. Since then, there followed many results of interpretations, embeddings, reconstructions, simulations, etc. among Tarskian and prooftheoretic logics. Such results have often been used to justify some statement of inclusion or relative expressiveness between the logics at issue.
As opposed to the case of modeltheoretic logics, until recently there was no attempt to give a precise definition of relative expressiveness in this framework. To the best of our knowledge, Mossakowski et al. [MDT09] were the first to give an explicit formal definition of translational expressiveness for logics, that is, an expressiveness relation based on the existence of certain kinds of formulamappings. We will expose their definition and show that it is still not adequate. Then, some adequacy criteria for expressiveness are proposed and a formal criterion for translational expressiveness is given.
Structure of the paper
This paper presents the following panorama on relative expressiveness between logics:

Relative expressiveness between logics (intuitive concept as given by (E))

Adequacy criteria for expressiveness

Approaches to (*) hopefully satisfying (a)
 [label=

singleclass
 [label=

formal proposals: ,


multiclass
 [label=

formal proposals: ,


translational
 [label=

formal proposals: Mossakowski et al.’s and .




In the framework of multiclass expressiveness will be presented and two formal criteria will be analysed, one from [GMV07] () and other from [Kui14] (). We argue that, using the intuitive explanation of expressiveness given above, there are counterexamples to both. In the sequence, we investigate what is wrong with them and propose that moving to an even wider framework, encompassing a greater range of logics and lacking structuremappings, might be promising.
In we present Mossakowski et al.’s formal criterion for translational expressiveness and show that, due to a result of [Jeř12], it is still not adequate. Then, some basic adequacy criteria for expressiveness will be proposed. In the sequence we analyse some formal conditions related to translations already appearing in the literature and investigate whether they satisfy the adequacy criteria. Finally, a formal sufficient condition for translational expressiveness () is proposed. We will argue that satisfies the criteria and is materially adequate.
2 Multiclass expressiveness
2.1 M. GarcíaMatos and J. Väänänen on sublogic
GarcíaMatos and Väänänen gave a multiclass definition of sublogic. Their definition is similar to one given in [Mes89] but is laxer.
Definition 2.2.
A logic is a sublogic of (in symbols ) if there are a sentence and functions , such that:

For every exists a such that and

For every and for every , if , then ( iff )
Thus, if the class of structures of a logic is richer than the class of structures of a logic , one could still allow a comparison between and , by restricting to the translatable structures, i.e. those which satisfy some condition and then use a function to translate this reduced class of structures into structures.
A problem with
Let be a trivial propositional logic in some given signature, and let () be the set of is truth tables together with a valuation. Let be any logic that has at least one valid sentence and let the formula of the definition above be such . Define the following mappings

. For every , .

. For every , .
Then it is easily seen that both items (a) and (b) above are satisfied.
Thus, according to this definition of sublogic, every logic containing at least one valid formula has a trivial sublogic. If we think on the usual meaning given to “sublogic”, this not plausible at all, since the logic could be nontrivial and might even lack a trivializing particle, so how come it could have a trivial sublogic?
It is not enough to require that the mapping be injective. Using an idea of [CCD09, p. 14], take for target logic any that has a denumerable number of valid formulas and define the mapping from the formulas of the trivial logic to formulas as . Still we have that has a trivial sublogic, once more, may be any logic with a denumerable number of validities, also lacking a trivializing particle.
Naturally, the usual senses of logic inclusion, that is, through language or axiomatic extensions do not apply here. The only way to make sense of this is to interpret the above cases as saying that a trivial logic can be simulated in any logic containing at least one validity. This capacity of simulating a logic is an expressive capacity, therefore the definition above is better seen as a definition of expressiveness. Yet, as an expressiveness relation, it is noteworthy that no restriction on the translation functions and are imposed, so one may wonder whether the definition overgenerates.
We are not in position to settle definitively this question. However we will give a plausibility argument to the effect that we should impose stricter conditions on model and formulamappings, since there is a natural and reasonable extension of the above definition that indeed overgenerates. Though not, strictly speaking, a counterexample, the case to be presented below shall give evidence that there is an intrinsic problem with the above proposal for multiclass expressiveness.
As we said, the sentence on the above definition of is intended to cut structures that are meaningless from the point of view of . Apparently, it would do no harm to the idea behind to allow to be a recursive set of sentences, as it is normally done in works dealing with translations of logics and conversion of structures (e.g. [Man96, p. 270]). This would be useful if the logics at issue have no conjunction, so that could be a finite set of sentences; or if the low expressive power of the logics and makes that the structures to be reduced into structures be only characterizable through an infinite but recursive set of sentences. This happens in the case of manysorted logic () and . If is not allowed to be an infinite set of sentences, then would not be a sublogic, in the above sense, of , which is implausible. Though the conversion of structures into structures is mentioned [GMV07, p. 23], the case of a given signature containing infinitelymany unary symbols is not considered. To convert structures into structures then one needs to make sure that unary predicates to be converted to manysorted domains are nonempty. This would only be accomplished by setting [Man96, p. 260].
However, if one allows such modification another implausible situation occurs. Consider the classical propositional logic () and a propositional logic , defined by Béziau [Béz99]. shares all the definitions of the classical propositional connectives, except for negation, where it has only one “half” of its clause: for a model and formula , if , then ; the converse direction does not hold.
Béziau shows that there is a translation from into . Below we will give Mossakowski et al.’s presentation of it, which includes also a model translation [MDT09, p. 107]. Given an nary connective , a translation is literal for if ; for an atomic formula , is literal when . Define the mapping ( as follows:


,

literal for and atomic formulas;


and

,

where comprises the truthtables for each connective and a valuation on the propositional variables. Notice that takes a model, keeps the valuation and replaces the truthtables for the corresponding ones.
Then we have that
Theorem 2.3 (Mossakowski et al.).
if and only if .
The model mapping is surjective, so that it obeys (a) above.
Now Mossakowski et al. (ibid, p. 100) define a mapping also from to using an auxiliary set of formulas constructed out of formulas.
Define the mapping as follows:


For every , , where is a propositional variable.

Define as the following set of formulas, for :




.
The purpose of is to encode the semantics of into the propositional variables , since every formula is translated into one of such , in a model satisfying the valuation of the propositional variables is forced to respect the semantics of . For example, in , if , then it holds that , but the converse direction does not hold. This is simulated in the models satisfying by the fourth clause above: if , then which implies that . But, as in , it does not hold that if , then .
Now define the modeltranslation :

Let be a model satisfying . Then is defined as follows:

For every formula , iff .

is also surjective (so it obeys (a) in the criterion for sublogic above). Then we have that
Theorem 2.4 (Mossakowski et al.).
iff and .
Therefore, by the above results and according to the extended definition of sublogic, we would have that and are one sublogic of another, which is not plausible. is not a sublogic of in the sense of language/axiomatic extension. Neither they are expressively equivalent, using above, since the “halfnegation” present in is not available in .
The problem is that the translation from to uses a trick to sneak in the semantics of into . Restricting the models that satisfy , one simulates the behaviour of formulas in the propositional variables and sustain such behaviour through the modeltranslation.
The modified version of , allowing to be a recursive set of sentences looks at least as “natural” as the original one. Even considering the original definition 2.2 we can see that there is something wrong with it, in not requiring any kind of preservation of the structure of formulas e.g. by forcing to be inductively defined through the formation of formulas. Then one may conjecture that, among more expressive logics, there be translations () where maps entire formulas to propositional variables and, with a sentence restricting the target structures, is able to mimic the semantic behavior of . Then it is very doubtful that the obtained would have the same meaning as .
Thus, we think we have good reasons to consider that GarcíaMatos and Väänänen’s definition of sublogic is not adequate. It would certainly be better to use a stronger notion of translation, paying attention to the structure of formulas. Only then the meaning of the targetformulas could be said to match the meaning of the sourceformulas. Below we will see that a development along this line appeared in the literature. Nevertheless, there is still a structureattentive translation that “cheats” similarly as the one above, mimicking the semantics of one logic into the other.
2.5 L. Kuijer on multiclass expressiveness
In his doctorate thesis [Kui14] Kuijer studies the expressiveness of various logics of knowledge and action, these logics are taken in the modeltheoretic sense. He notices that there are some results relating logics similarly as in singleclass expressiveness.
The purpose is to investigate features shared by all the results and construct a criterion, to be called “”, based on these features. Similarly with the work of GarcíaMatos and Väänänen exposed above, these prototypes involve translations of sentences and translations of structures. So a translation from to is a pair (), with and or , such that satisfies some given conditions.
A first plausible condition is that () must preserve and respect truth:
Definition 2.6 (Truth preserving).
A translation with and is truth preserving if, for every and
if and only if .
Then a tentative definition of could be
is at least as iff there is a that is truth preserving.
The problem is that the requirement of truth preservation is very weak, indeed there are several trivial truthpreserving translations among almost every logic. Kuijer gives the following example [Kui14, p. 88].
A trivial translation
Let be any logic on possible world semantics such that is countable and let be a logic where is a countable set of propositional variables but with no connectives and where is a class of models with possible worlds. Thus, every is a set of possible worlds with a valuation.
Define a truthpreserving translation () from to in the following way: map every to a propositional variable , maps a model to a model taking the set of possible worlds of and removing every other structure, and with the following valuation . Then clearly, by definition, is a truth preserving translation.
Defining
Since in the above example is an arbitrary logic on possible world models, if truth preservation were the only condition for multiclass expressiveness, would be at least as expressive as , which is absurd, given that has scarce expressive means. Nevertheless, truthpreservation is clearly a necessary condition. Thus, one must find other features a translation must satisfy in order to serve as a formal elucidation of the notion of multiclass expressiveness.
Another immediate criterion that comes to mind in order to avoid the trivial translations is to require the preservation of validities and entailment relations. However, some of the chosen prototypical translations do not preserve validity and some do not preserve entailment. Since the idea was to capture the essential features shared by all prototypical translations in , none of these can be imposed as a necessary condition.
Kuijer then goes through a number of tentative criteria, e.g. preservation of atomic formulas, of subformulas, etc., and shows that they are either too lax or too restrictive. Among the lax criteria, that is, the ones that are satisfied by some trivial translation, is one that Kuijer considers nonetheless important, the criterion of being model based:
Definition 2.7 (Model based).
A translation is model based if there are two functions such that, for all , we have that .
A model based translation would force to preserve some structure of and prevent that the pointed models and be translated to completely unrelated models.
Finally, the condition that apparently divides the good from bad translations and gives a reasonable notion of multiclass expressiveness is the criterion of being finitely generated. For the sake of simplicity, some aspects of the definition below are not completely formalized.
Definition 2.8 (Finitely Generated).
Let and be such that is generated by a set of propositional variables and a finite set of connectives, for . Let , then a translation is finitely generated if can be inductively defined by a finite number of clauses of the form
for
where is an sentence constructed out of and possibly containing , for ; and where is the range of the , e.g. if a given is to be replaced by a formula or only by an atomic formula.
The set contains the special propositional variables to be used in the translation clauses, for which one can substitute formulas. An example of such a translation clause is: for ; and for .
The idea is that (ibid, p. 110) it is the fact of being inductively defined and thus respecting (some) of the structure of the formulas that sets the finitely generated translations apart from the trivial translations. Thus Kuijer concludes that the truthpreserving translations giving rise to an expressiveness relation could be characterized as the ones being finitely generated and modelbased. Therefore, the final criterion given for multiclass expressiveness is (ibid, p. 111)
Definition 2.9 (Expressiveness).
Let and be such that is generated by a set of propositional variables and a finite set of connectives for .
Then is at least as as iff there is a translation from to that is model based, finitely generated and truth preserving.
A problem with
Kuijer had no pretensions that his multiclass definition were to be the generalization of expressiveness as given by the singleclass framework. The aim was to find only a “reasonable generalization” (ibid, p. 83). While keeping this in mind, we would like to argue that his proposal is still not good enough as a criterion for multiclass expressiveness. This is because one can find a pair of logics , such that is intuitively more expressive than , although is at least as as .
The logics at issue are Epstein’s relatedness logic () [Eps13, p. 80] and classical propositional logic (). The logic besides the truthfunctional connectives, has a relevant implication “”, which is the reason it is intuitively more expressive than , which lack such a connective. The referred translation would imply that is at least as as .
Despite the circumscribed character of Kuijer’s criterion, we think that a reasonable generalization of singleclass expressiveness should be able to deal with a reasonable amount of logics, not only with a handful of them. Particularly when the logics at issue are in the literature, and have not been constructed in an adhoc fashion just to give a counterexample. Finally, there is nothing specific about the logics appearing in the counterexample, so it is quite possible that there are also modal counterexamples.
Epstein presents with the connectives . The first two are defined as usual and the underlying idea for interpreting the relevant implication symbol “” is as follows. It holds that whenever materially implies and both are subjectmatter related to each other through a relation defined on all propositional variables. Specifically, for propositional variables and sentences and , holds if and only if for some occurring in , and occurring in , it holds that . Thus, the truth table for “” is the one for material implication with an additional column for , so that if holds and is true, then is true; else, if does not hold, then is false.
Let be a signature for . An model () is formed by the truthtables for , a symmetric and reflexive relation on formulas and a valuation . For propositional variables , let . Let be defined on (note we use here to emphasize that it is a material implication).
We will see below that there is a truthpreserving, modelbased and finitely generated translation . The mapping is defined as follows:


literal for and atomic formulas.
^{15}
Here the basic idea for the translation of comes from the definition of “”: materially implies and both formulas are related through . As the translation is defined inductively through the formation of formulas by a finite number of clauses, it is finitely generated.
Now, from an model (), one easily defines a transformation from models to models. Let , where, for take all the truthtables in , excluding the one for . Define as follows (adapted from [Eps13, p. 300]):

;

iff holds.
Clearly is modelbased.
Both and satisfy a semantic deduction theorem (ibid, p. 299). To prove that is truthpreserving, one has to prove only that, for an arbitrary model , it holds that
Theorem 2.10 (adapted from Epstein).
if and only if .
Corollary 2.11.
is at least as as .
The main question now is: does show that is at least as expressive as ? We do not think it is reasonable to say so, since the extra expressiveness brought about by the implication connective in is only by a trick mimicked in . Independently of the modeltranslation to give the intended truth values for the “relevancemimicking” variables , it is not possible to have a relevant conditional in , by say, adjoining to a conditional such variables . To do so, would require too much for the intended meaning of such variables. Surely this would not augment the expressive power of the propositional logic, as it concerns only an interpretation of propositional variables, and intuitively, specific interpretations of propositional variables do not influence the expressiveness of a logic.
Anyway, the modelmappings are not essential for these translations using indexed variables, they only facilitate their definition. An early example was given by Richard Statman in [Sta79] where a translation of into its implicational fragment is presented. There, the conjunctions are mapped to implications containing , among formulas of the sort , , etc. Here the situation is entirely different since the prooftheoretic behaviour of individual conjunctions are encoded in specific variables using implicational axioms.
Coming back to Kuijer’s criterion, we argued above that it is not enough to give an intuitively adequate account of expressiveness. If the model mapping were not from the source logic to the target logic but viceversa, then there would not be such truth preserving mappings from to , as there would be no way to construct the relatedness predicate out of a model. Kuijer discarded such a definition of the model mappings since it implies that any truthpreserving translation is also validity preserving,
Let us analyse a possible strengthening on the formula translation. We will not give a detailed analysis of features of translations since it suffices to notice that Epstein’s translation preserves completely the structure of the formulas, except for . For this case, additional propositional variables must be introduced to bear the intended meaning of (variables whose interpretation in is sustained by the model translation.) If one required that be compositional, that is, every ary connective of the source logic is translated by a schema of the target logic, then the above translation would not pass the test. This is because is translated through the schema , and by the schema . If the translation were compositional, dealing with the same connective, the same translation schema would be used.
The problem of adopting this criterion is that it implies that the connectives be translated one at a time, and again some of the paradigmatic translations selected by Kuijer takes into consideration sequences of connectives, so they would not satisfy it.
Therefore, to prevent translations such as those above from passing the test for multiclass, one would have to use a criterion for that is stronger than being finitely generated, but weaker than being compositional. Nevertheless, the enterprise of placing restrictions on the formula translations alone seems not to be promising, as the modeltranslations play a major role in the counterexamples presented above. On the other hand, placing also restrictions on modeltranslations and making them fit with the restrictions on formulatranslations is a very complex enterprise, and there may be better alternatives.
Given this situation, we would like to suggest a change of perspective as regards relative expressiveness between logics. Below, some comments will be made regarding the nature of the notion of expressiveness and its relation with the concept of logical system it applies to.
2.12 Singleclass expressiveness vs multiclass expressiveness vs translational expressiveness
Now we would like to make some remarks on the study of the relation of expressiveness between logics. As we commented before, in the singleclass framework it is very simple to define relative expressiveness, since there is a common ground, the structures, where one can compare whether the sentences have the same meaning. Now consider the multiclass framework, if and are defined on different classes of structures, how would we know whether an sentence and an sentence have the same meaning? After all, in this case it trivially holds that .
As we saw, for this task new tools are needed: a modelmapping or ;

,

,

,
^{18} 
.
Thus, now the weight goes on the notion of translation . As we saw in the examples presented above, for () in , and () in , the task of establishing the congruence between the pairs (, ) and (, ) by means of translations is very difficult. Basing it on satisfaction is far away from being sufficient, since we can easily devise translation functions such that satisfies iff satisfies .
On the other hand, imposing conditions on is a complex enterprise, because either it undergenerates or, by a little breach, it overgenerates. Moreover, the need to have modelmappings besides formulamappings may open up a back door to undesirable translations, to see it, consider again the examples offered against GarcíaMatos & Väänänen’s and Kuijer’s approaches. All of them use some “trick” in the formulatranslation function and sustain it through the modeltranslation. Then it is of little help to place structural restrictions on formulatranslations, as did Kuijer. He also tried placing restrictions on modeltranslations, but it did not help either.
Therefore, it might be more promising to move to a wider framework of relative expressiveness, dispensing with the semantic notions altogether. In this framework, to be called “translational expressiveness”, we would then concentrate the investigations on the conditions on formula translations. The aim is to find the set of conditions that better preserve/respect the theoremhood/consequence relation and the structure of formulas of each logic. This way a reasonable formal criterion of expressiveness for Tarskian and prooftheoretic logics (TPL, for short) would be obtained, and a bigger range of logics would be comparable. Finally, these advantages would arguably come at no cost, since this wider enterprise would not be more difficult than multiclass expressiveness.
The big difference between the approaches of expressiveness is not in the division between expressiveness for modeltheoretic logics and for TPL, but in the division, in modeltheoretic logics, of expressiveness within the same and within different classes of structures. Naturally the most direct concepts of expressiveness are linked with the capacity of characterizing structures, but this only applies when comparing the same class of structures.
If one allows translations between structures, such capacity is no longer at issue. Once we depart from the safe harbour of a single class of structures for comparing logics, then all bets are off. Multiclass expressiveness does not guarantee a firmer grasp of the intuitive concept of expressiveness anymore than translational expressiveness. Since the move to a wider framework might not only free us from problems inherent to multiclass expressiveness, but also allow a bigger range of comparison of logics, then the prospects for the enterprise are better.
As we said in the introduction, people have been using informally some concepts of translational expressiveness between logics. However, as opposed to what happens with modeltheoretic logics, to the best of our knowledge, in the literature there is only one explicit and formal criterion in this framework, that of [MDT09]. In the next section, their proposal will be analysed and we will show that it is not adequate. We shall then propose some adequacy criteria for expressiveness and a formal criterion in the framework of translational expressiveness will be given. We then argue that the criterion satisfies the adequacy criteria.
3 Translational expressiveness: obtaining a still wider notion of expressiveness
In this section we will deal with logics in the Tarskian and prooftheoretic sense. We also mention logics taken as a closed set of theorems/validities, to be called simply “formula logics”. Let and be logics, be a set of formulas and a translation mapping formulas into formulas in such a way that for each formula :
if and only if .
In this case is translatable into with respect to theoremhood.
If it is the case that
if and only if
then is translatable into with respect to derivability [PM68, p. 216]. The later translations are known as conservative translations [FD01].
Definition 3.1 (Conservative translation).
A conservative translation is a translation with respect to derivability.
Whenever we want to refer indistinctly to translations with respect to theoremhood or conservative translations, the term backandforth will be employed.
Definition 3.2 (Backandforth translation).
A translation is backandforth if it is either a theoremhood preserving or a conservative translation.
3.3 Mossakowski et. al.’s approach
As far as we know, Mossakowski et al. [MDT09] proposed the first explicit criterion for the concept of sublogic and expressiveness in the framework of translational expressiveness:
Definition 3.4 (Sublogic).
is a sublogic of if and only if there is an injective conservative translation from to ;
Definition 3.5 (Expressiveness).
is at most as expressive as iff there is a conservative translation .
The authors do not explain why sublogic requires injective conservative mappings while expressiveness does not. Anyway, we will see that these criteria for sublogic and expressiveness via conservative mappings do not work.
The conception that conservative translations could give rise to a notion of expressiveness and also a notion of logic inclusion has been supported more than once. For example, in [Con05, p. 233] it is said that the existence of a conservative translation (maybe injective or bijective) would give rise to some kind of logic inclusion between Tarskian logics.
Unfortunately, conservative translations will not make a reasonable concept neither of sublogic nor of expressiveness. Due to a result of Jeřábek [Jeř12], explaining expressiveness and sublogic through conservative translations would make include and be at least as expressive as many familiar logical systems, e.g. firstorder logic. He proved the following result (ibid, p. 668), where for a logic , a translation is most general whenever it is equivalent to a substitution instance of every other translation of to .
Theorem 3.6 (Jeřábek).
For every finitary deductive system over a countable set of formulas , there exists a conservative most general translation . If is decidable, then is computable.
The defined mapping is injective.
The author criticizes the notion of conservative translation for not requiring the preservation of neither the structure of the formulas nor the properties of the source logic [Jeř12, p. 666]. Thus, it must be strengthened in order to serve for an expressiveness measure. This could be done in a simpler way by requiring injective, surjective or bijective mappings. As Jeřábek’s mapping is injective, only requiring injectiveness will not do. As a matter of fact, it seems that already requiring injectiveness one is overshooting the mark. Since in this way would not be as expressive as . Any mapping would have to map both sentences and to the same sentence , so it would not be injective.
Other kinds of strengthening hinted by Jeřábek’s (ibid) are:

force the mappings to preserve more structure of the source logic sentences in the target logic;

force the mappings to preserve more properties of the source logic.
The adequacy criteria for expressiveness to be given below will require to some extent (1) and (2).
3.7 Adequacy criteria for expressiveness
As we saw above, Mossakowski et al. [MDT09] gave a proposal for a wide notion of expressiveness: by means of the existence of conservative translations. Due to Jeřábek’s results on the ubiquity on this kind of translation, their definition is not adequate. Maybe we should step back and think about some adequacy criteria every approach to expressiveness ought to accomplish.
The intuitive explanation for expressiveness given in the beginning elucidates relative expressiveness in terms of a certain congruence of meanings. It appears already in a more direct form in Wójcicki’s Theory of Logical Calculi [Wój88, p. 67], and we place it as the first adequacy criterion
[Adequacy Criterion 1] is at least as expressive as only if everything that can be said in terms of the connectives of can also be said in terms of the connectives of .
Here, for “being said in terms of the connectives” there can be stricter interpretations (as proposed by Wójcicki, Humberstone, Epstein) and wider interpretations (as proposed by Mossakowski et al. and us), to be developed below.
There are some metaproperties of logics that are intuitively known to limit or increase expressiveness. Thus, the presence/absence of such properties can be used to test whether there can be or not an expressiveness relation between the given logics. A first one coming to mind is that nothing can be expressed in a trivial logic, so it cannot be more expressive than any logic. Another one has to do with the relation between expressiveness and computational complexity. This relation has even been stated as the “Golden Rule of Logic” by van Benthem in [vB06, p. 119], where he says “gains in expressive power are lost in higher complexity”. Nevertheless, the “Golden Rule” is not quite useful here, since we know that in general neither a low expressiveness means low complexity,
Nevertheless the complexity levels of decidability/undecidability can be useful for expressiveness comparisons: if a logic is decidable, then it cannot describe Turing machines, Post’s normal systems, or semiThue systems. Therefore, a decidable logic cannot be more expressive than an undecidable logic , otherwise, would not be decidable!
The third metaproperty that could be useful when evaluating expressiveness relations (except, naturally, when dealing with formulalogics) is the deduction theorem. Though involved in many formulation issues, as we shall see, a logic has a deduction theorem when it has the capacity to express in the object language its deductibility relation. Thus, other things being equal, a logic having this capability is intuitively more expressive than another one lacking it. Therefore, it is desirable that an expressiveness relation carries with it the deduction theorem, so that (a) below apparently should hold
(a)
if is more expressive than , and has a deduction
theorem, then so does .
We have some issues here. Being formulation sensitive, it is complicated to define in which circumstances the existence of a deduction theorem for a logic implies its existence in another logic, whenever there is an expressiveness relation between them. For example, a less expressive logic might have the standard deduction theorem,
Cases like these constrain us to limit the role of the deduction theorem in expressiveness relations, admitting wider formulations of it. Thus we are forced to adapt (a) accordingly so as to be able to take into account such phenomena. Finally, we have the metaproperty related adequacy criterion.
[Adequacy Criterion 2] It cannot hold that be more expressive than when
is non trivial and is trivial;
is undecidable and is decidable;
satisfies the standard deduction theorem and the language fragment of purportedly as expressive as does not satisfy (not even) the general deduction theorem;
The last criterion reflects the intuition that expressiveness is a transitive relation and there are logics that are more expressive than others.
[Adequacy Criterion 3] (Taken from [Kui14]) The expressiveness relation should be a nontrivial preorder, that is, it should be a transitive and reflexive relation, and there must be some pair of logics and such that is not at least as expressive as .
We now analyse with greater detail the criteria 1 and 2.
Criterion 1 on “whatever can be said in terms of the connectives”
We can understand this criterion as saying “every connective of is definable in ”. But the usual notion of definability is either treated within the same logic, or between different logics within the same class of structures. As we intend to deal with translations between logics, the usual notion of definability is too rigid. We must give a broader reading of the criterion 1 in order to understand it as imposing an intuitive restriction on translations between logics. Thus the idea is to impose restrictions on translations so that
satisfies only if, intuitively, everything that can be said in terms of the connectives of can also be said in terms of the connectives of ; let us say in shorter terms that this happens only if the connectives of are generally preserved in .
In the sequence some candidates for such are listed, the backandforth condition was given before.
Definition 3.8 (Compositional).
A translation is compositional whenever for every ary connective of there is an formula such that .
Definition 3.9 (Grammatical).
A grammatical translation is a backandforth compositional translation such that, for a sentence , may contain no other formulas other than the ones appearing in , where appears in (thus, no parameters are allowed).
Definition 3.10 (Definitional).
A definitional translation is a grammatical translation for which for every atomic .
We have four proposals for filling the above list of restrictions. All of them require basically two conditions, taking as the backandforth condition. In decreasing order of strictness, there is divergence in taking as a
 [noitemsep]

definitional translation (Wójcicki and Humberstone),

grammatical translation (Epstein and apparently Koslow),

generalrecursive translation (to be defined below),

surjective conservative translation (Mossakowski et al.).
Humberstone [Hum05], recalling Wójciki’s definitional translations and intuitions about expressiveness, guessed that if there is a definitional translation between and , then all connectives in are preserved in .
For Epstein [Eps13, p. 302], a grammatical translation is a homomorphism between languages and thus it yields a translation of the connectives. The justification is that such translations are only possible when for each connective in the source logic, there corresponds a specific structure in the target logic that behaves similarly. Thus, through a grammatical translation, the connectives of the source logic are generally preserved in the target logic. Koslow [Kos15, p. 48] also allows that a connective from one logic “persists” in if there is a homomorphism from to .
According to Mossakowski et al. [MDT09], grammatical translations are too demanding for the task, as many useful and important translations are nongrammatical (e.g. the standard modal translation). For them, instead of seeking to preserve the structure of the formulas, it would be better to preserve the prooftheoretic behaviour of the connectives and to treat the connectives only as regards this behaviour (ibid, p. 100). In this paper, some prooftheoretic conditions on the connectives are listed, e.g. for conjuntction the condition is iff and . This formulation may lead one to think that here shall be a logical constant, and not possibly a formula (think of , where ); naturally in the first case, the whole proposal would make no sense. In table 1 we reformulate the conditions to reflect their proposal more clearly, where is an arbitrary formula that stands for the connective .
falsum  , for every 

conjunction  iff and 
disjunction  iff and 
implication  iff 
negation  iff . 
Definition 3.11 (presence of a prooftheoretic connective).
A prooftheoretic connective is present in a logic if it is possible to define the corresponding operations on sentences satisfying the conditions given in table 1.
We shall now investigate this idea in detail and argue that, as it is, the preservation of connectives would require mappings stricter than conservative translations otherwise the notion of the “presence” of a connective must be relaxed.
Drawbacks on the preservation of prooftheoretic connectives A translation transports a given connective if its presence in implies its presence in , the converse implication is called reflection [MDT09, p. 100]. It is claimed (ibid) that if a mapping is conservative and surjective, then all proof theoretic connectives of are transported to and all prooftheoretic connectives present in are reflected in . However, this claim must be taken with a grain of salt, let us see why.
Let be a logic having a prooftheoretic conjunction according with the table 1 above and suppose there is a surjective conservative mapping . For formulas , let be a set of formulas with and . Then it holds that
( and ) iff iff .
Thus, would have prooftheoretic conjunction. The grain of salt is that, once no structural restriction is imposed upon , it is not necessary that be constructed out of and . In this case, it seems at least unnatural to say that is an operation on the sentences and .
Therefore, we must relax what it means for a connective to be present in a logic. One has to say that e.g. the prooftheoretic conjunction is present in a logic if, for all formulas and set of formulas , there is a formula such that ( and ) iff . A similar reformulation should be given for the other connectives. In this case, though, whenever it holds that and , then any theorem in the place of serves to satisfy this condition for conjunction.
For example, take a Tarskian logic defined on the signature , where are propositional variables and the constant for logical truth. Then has prooftheoretic conjunction since and holds iff . This is probably unproblematic and a consequence of the meaning of . Nevertheless, for some cases this approach to the presence of connectives has some downsides. For example, restrict to the signature . Then has the prooftheoretic conditional, since it holds that

iff , iff , iff and iff .
But if the signature were incremented by another variable , then the resulting system would no longer have a prooftheoretic conditional, since for no it would hold that iff . This volatility of the presence of prooftheoretic connectives is unreasonable.
Recapitulating, the idea of this approach is that one shall define the mappings so as to preserve the prooftheoretic connectives, instead of requiring the mappings themselves to preserve the structure of the formulas. But if the mappings do not respect the structure of the formulas, what shall be called the presence of a connective, must also be relaxed.
Besides the inconvenients mentioned above, this proposal would be too restrictive in some cases. For example, Statman’s translation [Sta79] of into its implicational fragment shows how can one “express” (in some sense of the term) conjunctions using only implicational formulas; recent works have generalized this result so that any logic having a certain natural deduction formulation and having the subformula principle is translatable into the implicational fragment of minimal logic [Hae15].
Anyway, it must be borne in mind that to give a good and general definition of when a connective or operator is generally preserved is a difficult and spinous topic. Below we give another proposal, which is at the same time weaker (the translation mentioned above would enter) and stronger (requires structureattentive mappings).
Let us now consider the structureattentive translations and think on the minimum conditions on the preservation of the structure of formulas that would allow for a reasonable and general notion of preservation of connectives.
Generalrecursive translations: allowing contextsensitivity in a general preservation of connectives The criterion of compositionality given above a priori seems a reasonable condition for the preservation of connectives through translations. Notice that in the criterion the function that translates is the same that translates the subformulas . From this comes the compositionality: a translation of a formula is obtained through the same translation of its subformulas.
Thinking about the issue of translating a connective, it is also reasonable that the translation be sensitive to the context where the connective is inserted. This is the case in the translation in [DG00], where , but . Therefore, distinguishes between translating formula and formula, and this is done through the help of an auxiliary translation (see complete definition in section 3.27). Thus, there are translations between some logics where the mappings must be contextsensitive, so as to convey the proper meaning of some source connectives in the target logic.
There are also those cases where the connectives can be dealt contextindependently but auxiliary translations are needed anyway. The standard translation of modal logic to , besides some parameters, needs auxiliary translations for each formula of modal degree e.g. as but .
For the sake of simplicity, we will restrict our notion of contextsensitivity to whether or not the connective to be translated is in the scope of an unary operator. When the translation of a ary connective is sensitive as to whether it is on the scope of an unary , a simple solution is to treat as a composite ary connective to be translated. With the aim of capturing these cases, let us consider a sufficiently general kind of translation.
French in [Fre10] presents a concept of recursively interdependent translation that includes noncompositional translations that are still defined recursively through the formation of formulas. A generalization of his concept will be employed here, since the original has an unmotivated restriction allowing only unary auxiliary mappings. The generalization allows auxiliary mappings of any arity and also has a simpler notation. Let and be logics,
Definition 3.12 (GeneralRecursive).
Let be auxiliary mappings of any arity defined inductively on formulas. A translation from to is generalrecursive if, for every ary connective and formulas